Optimization, in general, means working out the values of a set of variables that returns the stated extremum of the objective function while satisfying the constraints imposed over the variables. Optimization techniques are applied to many different areas, including finance.

In finance, the objective is to find the optimizing values of the variables to have the highest expected return and lowest risk. Portfolio management is a fundamental activity of all economic agents.

The optimization problem of the portfolio manager can be expressed in two equivalent ways: the investor, assumed to be constantly rational in making decisions, is supposed to find the greatest expected return portfolio with the given risk or the lowest risk portfolio with the given expected return. These two problems are called duals and yield exactly the same solution set of variables.

The optimization problem has many different forms: the objective may be minimization or maximization, the constraints may be linear or nonlinear, the constraint may be “less than” or “greater than” type, etc. the following formulation of the problem can be manipulated to include all cases:

In this formulation F is the objective function and c(x) ≤ x is the set of constraints. If a x portfolio is selected in case there is another portfolio with a greater return and the same level of risk, then there is inefficiency. The set of all efficient portfolios (portfolios that have the lowest risk for any given return) constitutes the efficient frontier. Portfolios off this frontier should not be considered for investment.

Markowitz (1952) pioneered the study of the portfolio optimization problem and developed the “mean–variance approach” with the main assumption of normality. He was awarded the Nobel Prize in 1990 for his contribution to finance theory. this approach is still very popular and is applied by financial institutions.

Although Markowitz’s analysis was remarkable, it is being criticized because of being static and the unrealistic assumption of normality. The investor is not given a chance to update the portfolio until the end of the period, which is not realistic. This unrealistic assumption should lead to the opportunity cost of the better strategies possible.

The investor requires a model for such shifts of portfolios since the volatility of prices is high and the conditions are changing through time that requires working out the portfolio optimization problem on a continuous basis. At least Markowitz’s original work should be expanded to handle multiperiods. Another Nobel Laureate Merton (1971) recognized this and updated the optimization problem to the continuous case.

The optimization problem of hedge funds is an extension of the portfolio optimization. What is specific to this optimization problem is the expression of risk. For instance, Favre and Galeano (2002) achieve the mean-modified value-at-risk optimization with hedge funds. Duarte (1999) includes a short list of measures for risk and claimed that Markowitz’s study is a special case of his work.

The list constitutes mean variance (MV), mean semivariance (MSV), mean downside risk (MDR), mean absolute deviation (MAD), mean absolute semideviation (MASD), and mean absolute downside risk (MADR). There are several online Internet services that help investors find the best portfolio, based on their preference of risk measure.